MATHEMATICIANS OF HISTORY

Stephen Slattery • Tom Boffeli • Sam Gaspar • Michael Drees

RENE DESCARTES 31 March 1596 – 11 February 1650 Descartes was born in Touraine, France, but

spent most of his life in the Dutch Republic Also called the ‘Father of Modern Philosophy’ Best known for his work in philosophy, but his

discoveries in mathematics were equally astounding

Wrote dozens of works on anatomy, philosophy, mathematics, music, and also some novels

“I think, therefor I am” – Descartes most famous quote, referring to the existence of the human mind

DESCARTES’ ACCOMPLISHMENTS Developed the basis for modern calculus, leading to further

developments by Newton and Leibniz Fun fact: Descartes’ discovery of calculus was actually just for an example he used

in an unrelated work

Created a method to determine the number of positive and negative roots in a polynomial

Invented analytic geometry, or geometry using a Cartesian plane Discovered a version of the law of conservation of momentum Contributed greatly to the field of optics by discovering the law of

reflection and proving various optic-related problems Invented Cartesian geometry, which describes geometry with

algebra Began the use of superscripts to describe a number raised to a

power

LEONHARD EULER 15 April 1707 – 18 September 1783 Born in Basel, Switzerland, but spent most of his

life in Russia and Prussia Leader of his fields in math and science during his

lifetime Famous for his work in mathematics, mechanics,

fluid dynamics, optics, and astronomy His works, collectively, would be several hundred

pages He was at one time featured on the Swiss 10-franc

banknote “Madam, I have come from a country where

people are hanged if they talk”

EULER’S ACCOMPLISHMENTS

Provided major contributions in multiple of fields, including geometry, infinitesimal calculus, trigonometry, algebra, number theory, continuum physics, and lunar theory

Began the use of f(x) to describe the function “f” applied to “x” His work on the use of power series (expressing functions as sums

of an infinite number of terms) contributed to infinitesimal calculus Proved the inverse tangent function Began the use of exponential function and logarithms in analytic

proofs Created “Euler’s Identity,” a formula used to describe the relation

between the exponential function of a complex number to trigonometric functions

PIERRE DE FERMAT 17 August 1601 – 12 January 1665 Born in Beaumont-de-Lomagne, France, was

considered an amateur mathematician during his time

Attended the University of Toulouse, then the University of Orleans, and finally settled in Toulouse for the rest of his life

Known for his work in analytic geometry, differential and infinitesimal calculus, number theory, and probability

Creator of “Fermat’s Last Theorem” “I have found a very great number of

exceedingly beautiful theorems”

wired.com

FERMAT’S ACCOMPLISHMENTS

His work in Ad Locos Planos et Solidos was revolutionary in the field of analytic geometry

Invented a method for finding the maxima, minima, and tangents to various curves, leading to his work in quadrature

First person to examine the integral of general power functions, which he reduced to geometric series’

This formula assisted Newton and Leibniz in their study of calculus

Invented “Fermat’s factorization method” and “Fermat’s Last Theorem”

Fermat’s Last Theorem would not be proved until 1995 by Andrew Wiles

Claimed to have proved all of his own arithmetic theorems Developed the two-square theorem and the polygonal number

theorem

LEONARDO FIBONACCI c. 1170 – c. 1250 Born in Italy to Guglielmo Fibonacci Grew up with using Roman numerals, before realizing

that the Arabic number system was much more efficient Travelled around the Mediterranean to be taught by

Arabic mathematicians Famous for his creation of the Fibonacci Sequence, a

sequence with uncountable, amazing properties Compiled much of his work in his books, such as Liber

Abaci No words of Fibonacci’s have ever been recorded,

except those in his books

FIBONACCI’S ACCOMPLISHMENTS Introduced Arabic numerals in his work Liber Abaci

Became very popular in Europe, greatly effecting European thought

Fibonacci’s “claim-to-fame” is the Fibonacci Sequence Each number is the sum of the previous two numbers, starting with

0 and 1 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 …

The further the sequence proceeds, the closer the difference between consecutive numbers comes to the “golden ratio,” or 1 : 1.618

The golden ratio is known for its various mysterious properties, such as those related to ratio’s in nature

ÉVARISTE GALOIS 25 October 1811 – 31 May 1832 Born in Bourg-la-Reine, France, and performed

all of his mathematical work as a teenager Radical Republican during French monarchy,

which lead to his death in a duel Was schooled at Lycee Louis-le-Grand, where he became

bored of all other subjects except mathematics Published a number of papers on mathematics during his

teenage years Creator of the “Galois Theory” “The most worthwhile scientific books are those in which

the author clearly indicates what he does not know; for an author most hurts his readers by concealing difficulties”

GLAOIS’ ACCOMPLISHMENTS

Published a paper on continued fractions at age 17 Made a number of fundamental discoveries

pertaining to polynomial equations He submitted another two papers on this topic

Published three papers that were the basis for the Galois Theory

Described the concept of a finite field Did all of his mathematical work while suffering from

the suicide of his father and distracted by political activities

KARL FRIEDRICH GAUSS 30 April 1777 – 23 February 1855 Born in Lower Saxony, Germany (then part of

the Holy Roman Empire) Often called the “Prince of Mathematicians” Great influence in many fields of science and

mathematics such as number theory, statistics, geophysics, and astronomy

Known as a constant perfectionist and a hard worker, but did not publish many works

“I have had my results for a long time: but I do not yet know how I am to arrive at them”

Wjmc.blogspot.com

GAUSS’ ACCOMPLISHMENTS Wrote Disquisitines Arithmeticae, referred to as one of

the most influential mathematical works Proved the law of quadratic reciprocity Developed the method of “least squares fitting”

Which he never published, and another mathematician later took credit

Proved that every number is the sum of (at most) three triangular numbers

Developed the algebra of congruence's Calculated the orbit of the dwarf-planet Ceres

HYPATIA OF ALEXANDRIA c. AD 360 – March 415 Considered the first notable female mathematician Taught philosophy and astronomy in Alexandria,

Egypt Received much abuse throughout her life Edited and commentated a number of

mathematical works Also did work in astronomy Assassinated by an angry mob "Reserve your right to think, for even to think

wrongly is better than not to think at all"

Apod.nasa.gov

HYPATIA’S ACCOMPLISHMENTS

Provided commentary on a number of works Arithmetica by Diophantus Conics by Apollonius

Edited a number of famous works Almagest by Ptolemy Elements by Euclid

Wrote The Astronomical Canon Charted various celestial bodies Invented the hydrometer, used to determine relative

density of liquids

GEORG CANTOR 3 March 1845 – 6 January 1918 Born in Russia, but spent the majority of his

adult life in Germany Attended Federal Polytechnic Institute in

Zurich and the University of Gottingen Given a PhD for his thesis on number theory Most famous for his work on the set theory “A set is a Many that allows itself to be

thought of as One”

CANTOR’S ACCOMPLISHMENTS Developed the origins of the set theory, including both

finite and infinite sets Set Theory is the basis of modern analysis

Performed work on the theory of transfinite numbers Provided a definition for irrational numbers His approach to the idea of infinite revolutionized

mathematics by challenging the foundations of mathematical theories

Found a 1-to-1 correspondence between the points on a unit line segment and all of the points in an n-dimensional space

JOHANNES KEPLER Dec 27, 1571 - Nov 15,1630 Helped grandparents run an inn Sought to be ordained a priest Interested in astronomy and music Believed that God, the universe, and God are all

connected Succeeded Tycho Brahe as the Imperial

Mathematician Wrote several books including The Harmony of the

World and The Mystery of the Cosmos “I much prefer the sharpest criticism of a single

intelligent man to the thoughtless approval of the masses”

KEPLER’S ACCOMPLISHMENTS First to refer to the moon as a “satellite” First to explain the alignment of planets in the

Copernican system using geometry Defined the orbit of Mars as an ellipse Made advancements in optics and lenses Designed a telescope with two convex lenses Found the diameter of the moon Studied volume of solids Lead the way for infinitesimal calculus

FELIX KLEIN

Apr 25,1849–June 22,1925 Studied at University of Bonn Dreamed of becoming a physicist Co-wrote a book on line geometry Chairman of International Commission on Mathematical

Instruction “Mathematics has been most advanced by those who

distinguished themselves by intuition.”

KLEIN’S ACCOMPLISHMENTS Found connection between Euclidean and Non-Euclidean

geometry Discovered asymptotic lines and W-curves Studied “transformations” and their effect on geometry Applied fluid dynamics to function theory Interested in equations to the fourth or fifth degree Wrote about the icosahedron Used automorphic functions to connect algebra &

geometry Created a cylinder that can only be made in Non-

Euclidean space

GOTTFRIED WILHELM LEIBNIZ July 1,1646 – Nov 14, 1716 Sought to improve Aristotle's theory of categorizing knowledge Studied physics, theology, law, and philosophy Established schools of science all over Europe Served as an elected member of the Royal Society of London “The ultimate reason of things must lie in a necessary substance,

in which the differentiation of the changes only exists eminently as in their source; and this is what we call God.”

LEIBNIZ’S ACCOMPLISHMENTS Made calculations in calculus easier by developing a certain

notation Sought to coordinate the work of two societies to from one

research force Theorized abstract ideas of motion Dreamed of creating an advanced calculating machine Wrote many books including the “Dissertation on the

Combinatorial Art” Set rules for differentiating a function of a function. Was among the first to invent wind powered and water powered

pumps. Focused on the arithmetic binary system

NIKOLAI LOBACHEVSKY

Dec 1 1792 – Feb 24 1856 Originally planned to study medicine but later

focused more on mathematics and physics Born and raised in Russia near Siberia Well known for his ability to teach and give lectures Named rector of the University of Kazan Fell ill when he became overwhelmed with his work load “There is no branch of mathematics, however abstract,

which may not some day be applied to phenomena of the real world.”

LOBACHEVSKY’S ACCOMPLISHMENTS

Made strides in mechanics, hydrodynamics, integration Incorporated differential equations, the calculus of variations, and

mathematical physics into standard mathematics Sought to create a geometry in which Euclid’s fifth postulate does

not necessarily hold true. In this way, he created a more general geometry that included the

specifics of Euclidean geometry Published works on hyperbolic geometry and non-Euclidean

geometry Learned to easily approximate the roots of algebraic equations. Solved algebraic equations the way a present day computer would Established rules for the foundations of geometry

MARIN MERSENNE Sept 8 1588 – Sept 1 1648 Attended school with Rene Descartes Joined the Order of the Minims because

he was dedicated to prayer, study, and scholarship Publically argued against atheism, witchcraft, and

sorcerers Was known as a devout defender of Aristotle “(Animals) have no enlightenment except what they

must have to take their nourishment and to serve us for the uses to which God has destined them.”

MERSENNE’S ACCOMPLISHMENTS As a musician, he was interested in acoustics and the speed of

sound Investigated specifically the frequency, length, and weight of

vibrating strings Lead the way for the investigation of large prime numbers Studied the law of motion for falling bodies. Utilized combinations and permutations in mathematical

calculations Conducted experiments with a scientific barometer Calculated the density of water to be 19 times greater than air Was so dedicated to science that he willed his body to be used

for biological research.

JOHN NAPIER 1550 – April 4th 1617 Studied at St. Andrew’s University and possibly at

University of Paris, in Italy, or in Netherlands Described as a fanatical and fervent Protestant Authored Plaine Discovery of the Whole Revelation of

St. John and Mirifici Logarithmorum Canonis Descriptio Said to be “in league with the powers of darkness” “There is nothing that is so troublesome to

mathematical practice than the multiplications, divisions, square and cubical extractions of great numbers”

NAPIER’S ACCOMPLISHMENTS Created ivory numbering rods to simplify calculations Found a way to solve spherical triangles Easily multiplied, divided square roots and cube roots Wrote his logarithms with a constant of 107

In his calculations, used a base of 10 and a log1 = 0, Aided Kepler, who then lead to Newton's theory of

gravitation Logarithms and calculators greatly improved astronomy Practiced farm techniques such as using salts to manure

farm fields

SIR ISAAC NEWTON Jan 4 1643 - Mar 31 1727 Grew up in a difficult family environment (having to move

between parents, step-parents, and uncles) dealing with the death of several family members

Studied philosophy and astronomy but very little mathematics Newton was elected a fellow of the Royal Society and was later

named its president Gave up research and became Warden of the Royal Mint Authored Philosophical Transactions of the Royal Society. Vehemently fought to end counterfeiting of coins “If I have seen further than others, it is by standing upon

the shoulders of giants.”

NEWTON’S ACCOMPLISHMENTS

Biggest advancement was the theory of universal gravitation. Made strides in physics and mechanics of celestial bodies Used mathematics to connect things in orbit, projectiles,

pendulums, and free-fall near the Earth. Said that the pull of the Sun affected comets, tides and the lunar

cycles Developed method of fluxions found use for tangents, the lengths

of curves and functions Noted the spectrum of colors formed by a prism and theorized

about white light Set rules for differential and integral calculus Invented the reflecting telescope

ZENO OF ELEA• 490BC - 430BC• Attended Eleatic School• Philosopher• Referenced by Plato• Attempted to kill the tyrant

Demylus• “With his own teeth bit off

his tongue, he spit it in the tyrant’s face”

• Main accomplishment - Zeno’s Paradoxes

CHARLES BABBAGE 1791-1871 England From Peterhouse College Wrote history of calculus Part of London’s Royal Society

“A tool is usually more simple than a machine; it is generally used with the hand, whilst a machine is frequently moved by animal or steam power.”

BABBAGE’S ACCOMPLISHMENTS Part of the Royal Astronomy Society Much work on Calculus Wrote Memoires of Analytical Society, a

famous book on the history of calculus Engines Develeoped the small difference engine Translated many math books to English Knowm for differential and analytical calculus A famous math professor at Cambridge

MOHAMMED AL-KHOWARIZMI Born: 780 A.D Died: 850 A.D He was Persian, from Baghdad Also an astronomer and geographer Very little is known about his life outside of mathematics Known as the Father of Algebra“When I consider what people generally want in calculating, I found that it always is a

number. I also observed that every number is composed of units, and that any number may be divided into units. Moreover, I found that every number which may be expressed from one to ten, surpasses the preceding by one unit: afterwards the ten is doubled or tripled just as before the units were: thus arise twenty, thirty, etc. until a hundred: then the hundred is doubled and tripled in the same manner as the units and the tens, up to a thousand; ... so forth to the utmost limit of numeration.”

AL-KHOWARIZMI’S ACCOMPLISHMENTS

developed the concept of the algorithm in mathematics The word “algebra” comes from the name of one of his books,

Hisab al-jabr wa al-muqabala He synthesized much of the knowledge in mathematics from

the Greeks, Indians, and many other sources He created the place-marker symbol of zero which originates

in India Also, he did work on many other fields of math; trigonometric

functions, refinements in the geometric representation of conic sections, and aspects of the calculus of two errors

The use of Arabic numerals in mathematics is because of him He invented a systematic and logical approach to solving

linear and quadratic equations The math textbook that he wrote was the standard textbook

used at Europe’s universities during the 1600’s; therefore, he contributed to the education of many future mathematicians

MARIA AGNESI Born: May 16, 1718 Died: January 9, 1799 Born in Milan, Italy Very wealthy Lived during Renaissance Wrote the first book discussing both

integral and differential calculus Left mathematics at an early age

"I will finish the Instituzioni with a warning. The expert analyst should be industrious in trying to search for solutions to these problems and will be much more advanced by means of the techniques that are "born" during this process.”

AGNESI’S ACCOMPLISHMENTS Wrote Analytical Institutions-it was groundbreaking material on

integral and differential calculus Provides information on the analysis of finite quantities She provided elementary of maxima and minima, points of

inflection, and tangents Her best work is her work on the cubic curve whose equation is

x^2y = a^2(a-y), the Curve of Agnesi. Although she makes no claim to any original math

accomplishments, she played an important role of organizing the discoveries of many 18th century mathematicians, including Newton.

She discusses the analysis of infinitely small quantities In her work with integral calculus, she specifies rules for integration Dealt with the inverse method of tangents and differential

equations Discovered rules and equations to find the point of inflection

BERNOULLI FAMILY

Mid 1600s-late 1700s

Switzerland

One of the most successful

mathematical families of all time

Johann and Jacob we’re the most

dynamic members

“It would be better for the true physics if there were no mathematicians on earth.” Daniel Bernoulli

FAMILY ACCOMPLISHMENTS The problem of the catenary y=x^2 the function General method to solving a curve Hydrodynamica-a physics book on water Dividing a triangle into 4 equal sections with 2 parallel

lines Worked on curves and probability Daniel developed Bernoulli’s Principle The St. Petersburg paradox was developed by Daniel

JANOS BOLYAI

1802-1860 Hungary Known for non-Euclidean geometry Went to college in Vienna Left enormous amounts of math information after death“Out of nothing I have created a strange new

universe.”

JANOS BOLYAI ACCOMPLISHMENTS Famous for non-Euclidean geometry Worked on postulates, axioms Complex and real numbers Left 20,000 pages of mathematical

manuscripts, gave us much needed information on non-Euclidean geometry

Set up his own definition of a parallel Wrote his book: Geomtrical Examinations Wrote a paper called Respansio Hyperbolic geometry

GEORGE BOOLE

1816-1864 England Member of the Royal Society Boolean Logic Founder of computer science

“Of the many forms of false culture, a premature converse with abstractions is perhaps the most likely to prove fatal to the growth of a masculine vigor of intellect.”

Wehner.org

GEORGE BOOLE

Came up with Boolean Logic He is central to all computer science Logic Treatise on Differential Equations Taught math at Queens College Wrote many papers on probability Propositions could be used to find

equations

BRAHMA-GUPTA 598-670 India Wrote many books on astronomy

and math First person to call zero a number Did much work in liner equations

and series

"Oh Archimedes how come you did not discover this, If you had world would be much more advanced"

ANDREW WILES

• April 11, 1953 – Present• Awarded a PhD. in Mathematics

in 1980• Had a childhood dream to prove

Fermat’s Last Theorem• Asteroid 9999 Wiles is named

for him• ... I was a ten year old and one

day I happened to be looking in my local public library and I found a book on maths and it told a bit about the history of this problem and I, a ten year old, could understand it.

ANDREW WILES’ ACCOMPLISHMENTS

Proving the Taniyama-Shimura Conjecture for semi-stable elliptic curves

Proving Fermat’s Last Theorem Fermat Prize 1995 Wolf Prize 1995/6 Royal Medal 1996 IMU Silver Plaque 1998

JOHN VON NEUMANN

December 28, 1903 – February 8, 1957

Born in Budapest, Austria-Hungary

Showed incredible memory skills as a child

PhD in Mathematics at age 22

JOHN VON NEUMANN’S ACCOMPLISHMENTS

• Von Neumann Equation• Von Neumann Algebra • Artificial Viscosity• Backwards Induction• Game Theory• Von Neumann’s Inequality• Minimax Theorem• Von Neumann Universe

THALES 624BC-547BC Present Day Turkey Business man Predicted a solar

eclipse accurately Bought all of the olive

mills after he predicted a good harvest season

“Space is the greatest thing, it contains all things.”

THALES’ ACCOMPLISHMENTS

Water is the Physis Thales’ Theorem Another Theorem known as Thales’ Theorem

that deals with circles inscribed in circles Noted as the first Greek Philosopher by Aristotle Convinced a young Pythagoras to travel to

Egypt to further his education

GF BERNHARD RIEMANN

• September 17, 1826 – July 20, 1866

• Born in present day Germany

• Very timid at a young age

• Very interesting in Mathematics at a young age

• Doctoral Advisor - Gauss

BERNHARD RIEMANN’S ACCOMPLISHMENTS

Riemann Geometry Riemann-Roch Theorem Cauchy-Riemann Equations Grand Riemann Hypothesis Riemann Function LaRouche-Riemann Method Riemann Sphere

SRINIVASA RAMANUJAN

• December 22, 1887 – April 26, 1920

• Did not like school• By age 11, he had exhausted

the mathematical knowledge of 2 college students living at his house

• Mastered Trigonometry at age 13

• Married a 9 year old bride at the age of 26

SRINIVASA RAMANUJAN’S ACCOMPLISHMENTS

Ramanujan Prime Mock Theta Functions Ramanujan’s sum Landau-Ramanujan Constant Ramanujan Conjecture Ramanujan Theta Function

BLAISE PASCAL

June 19, 1623 – August 19, 1662

French Was allowed to watch and

study some of the greatest Mathematicians in Europe

Home taught by Father

BLAISE PASCAL’S ACCOMPLISHMENTS Pascal’s Triangle Pascaline(Pascal’s Calculator) Of the Geometrical Spirit Primitive Roulette Wheel The Provincial Letters The Pensées

AHMES

1680BC-1620BC Egyptian First known contributor

to Mathematics Scribe to the Rhind

Papyrus Showed division of 2 by

odd numbers from 3 to 101

msnucleus.org

QUIZ

Kepler – inherited the title of “Imperial Mathematician Klien – constructed a cylinder that cannot be made in

Euclidean space Leibniz – attmpted but failde to creat wind-powered

pumps Lobachevsky – Worked extensively in hyperbolic and

Non-Euclidean geometry Mersenne – Member of the Order of Minims Napier – Created ivory numbering rods Newton – Was the Warden of the Royal Mint

QUIZ QUESTION #1

Q: Who was the first known contributor to mathematics?

A: Ahmes

QUIZ QUESTION #2

Q: Who invented the pascaline?

A: Blaise Pascal

QUIZ QUESTION #3

Q: Which mathematician was made famous for his work on the Mock Theta Functions

A: Srinivasa Ramanujan

QUIZ QUESTION #4

Q: Fill in the blank…

LaRouche-______ Method (or)________ Sphere

A: Bernhard Riemann

QUIZ QUESTION #5

Q: Which Turkish mathematician from ancient times predicted a solar eclipse accurately?

A: Thales

QUIZ QUESTION #6

Q: Which mathematician is known for his work on Backwards Induction and Game Theory?

A: John von Neumann

QUIZ QUESTION #7

Q: Who solved Fermat’s Last Theorem?

A: Andrew Wiles

QUIZ QUESTION #8

Q: Fill in the blank…

_____’s Paradoxes

A: Zeno

QUIZ QUESTION #9

Q: Who developed the theory of universal gravitation?

A: Sir Isaac Newton

QUIZ QUESTION #10

Q: Who thought of the idea to use ivory numbering rods to simplify calculations?

A: John Napier

QUIZ QUESTION #11

Q: Who studied the law of motion for falling bodies?

A: Marin Mersenne

QUIZ QUESTION #12

Q: Which famous Russian scientist worked so hard that he became ill?

A: Nikolai Lobachevsky

QUIZ QUESTION #13

Q: Which mathematician sought to improve Aristotle’s theory of categorizing knowledge?

A: Gottfried Leibniz

QUIZ QUESTION #14

Q: Who found the connection between Euclidean and non-Euclidian geometry?

A: Felix Klein

QUIZ QUESTION #15

Q: Who defined the orbit of Mars as an ellipse?

A: Johannes Kepler

QUIZ QUESTION #16

Q: Who developed the origins of Set Theory?

A: Georg Cantor

QUIZ QUESTION #17

Q: Who was the first well-known female mathematician?

A: Hypatia

QUIZ QUESTION #18

Q: Who was often called the “Prince of Mathematicians” ?

A: Karl Guass

QUIZ QUESTION #19

Q: What famous French mathematician was a child-genius, living to only 21, and described the concept of finite field?

A: Evariste Galois

QUIZ QUESTION #20

Q: Who created a sequence where each element is defined by the sum of the previous two elements?

A: Fibonacci

QUIZ QUESTION #21

Q: Who created a theorem that would remain unsolved until 1995?

A: Pierre de Fermat

QUIZ QUESTION #22

Q: What famous Swiss mathematician was featured on the Swiss 10-france banknote?

A: Leonhard Euler

QUIZ QUESTION #23

Q: Who said the famous quote, “I think, therefore I am” ?

A: Rene Descartes

QUIZ QUESTION #24

Q: Who wrote the first book discussing both integral and differential calculus?

A: Maria Agnesi

QUIZ QUESTION #25

Q: Who was known as the “father of algebra”?

A: Mohammed Al-Khowarizmi

QUIZ QUESTION #26

Q:Who wrote History of Calculus?

A: Charles Babbage

QUIZ QUESTION #27

Q: What was the most successful mathematical family of all time?

A: The Bernoulli Family

QUIZ QUESTION #29

Q: Who created his own definition of the term “parallel” and wrote a book called Geometrical Examinations?

A: Janos Bolyai

QUIZ QUESTION #30

Q: Who was the founder of computer science?

A: George Boole

QUIZ QUESTION #31

Q: What famous Indian mathematician wrote many books on astronomy and math in the 600s?

A: Brahma Gupta

QUIZ QUESTION #32

Q: Who proved the law of quadratic reciprocity?

A: Karl Gauss

THANK YOU!